Friedmann-Robertson-Walker Spacetimes

Which solutions of Einstein's
gravitational field equations can accommodate a universe with a Hubble
expansion? Or, more figuratively, which pages in Einstein's great book might
belong to our universe?

The answer lies in a class of solutions of Einstein's
equations picked out a few simple conditions, Friedmann-Robertson-Walker spacetimes. These spacetime
can be sliced up into spaces that evolve into each other over time.

(It isn't automatic that a spacetime
can be cut up into nice spatial slices. A Goedel universe cannot be sliced up
nicely into spaces that evolve into each other over time.)

What character should the spaces have? Our observations of
our cosmos tell us that on the largest scale space is homogeneous and
isotropic. So we ask that the solutions have a homogeneous, isotropic space--that is a space that is the
same in every place (homogeneous) and in all directions (isotropic); and that
this space simply evolves with time.

This condition that space is homogeneous and isotropic on the
largest scale has been called the "cosmological
principle." That strikes me as a little dangerous. Naming the condition
the cosmological principle does no harm, of course, as long as we realize that
it is just a name. However, when the term "principle" is used, it is easy to
get the impression that the condition is somehow unchallengeable. That is
risky. Whether the universe is roughly homogeneous and isotropic is something
that should be determined by observation. It should not be elevated to apriori
heights.

The condition that the space is homogeneous and isotropic
restricts it to three general possibilities. Such a space must have constant
spatial curvature. We know from earlier that the three possibilities are:

Spherical
Positive curvature

Flat, Euclidean
Zero Curvature

Hyperbolic
Negative Curvature

A space of one of these three types will be the instantaneous
snapshots that comprise the "now" of the cosmology.

Each of these snapshots of space will be filled with a uniform matter distribution. Its composition is not
fixed. It may be ordinary matter, such as comprise planets and stars; or it may
be radiation; or it may be a mixture of the two. At present we have a mixture
that is heavily skewed towards ordinary matter. In the past, radiation was
dominant.

Finally the spaces of the cosmology cannot remain static.
They are either expanding or contracting. The first
case of expansion is the one that interests us most since it is what we
observe. As time passes, the space expands, its curvature, if it has any
decreases, and the distance between the galaxies increases. The figure shows
the worldlines of the galaxies with the spatial slices.

The most interesting feature of this figure is what happens
when we project the worldlines of the galaxies into the past. The galaxies get
closer and closer. Eventually, they converge onto a state of infinite curvature
and density. This is the initial state--the so-called "big
bang."

What the Big Bang really is

It is easy to misunderstand the nature of the big bang and
the expansion of the universe.

The popular image called to mind by the name big bang is
something like this. There is a huge empty space, with an infinitely dense
nugget of matter containing all future matter of the universe. At the moment of
the big bang, This nugget explodes. Fragments of this primeval nugget are
scattered into space, progressively filling it with an expanding cloud of
matter. This is NOT the modern big bang model.

Rather the expansion is the expansion of
space itself. The most helpful picture is of the rubber surface of a
balloon expanding. The galaxies are like dots drawn on the surface. They move
as the rubber sheet stretches. The galaxies fly apart because space expands. At
any instant, space is always full of matter; there is no island of fragments
expanding into an empty space.

Here's how one fixed volume of space will look over the course of cosmic evolution.

If we now project back to the big bang, we project back to a
time at which all matter and space were somehow compressed into a state of
infinite density. Einstein's gravitational field equations tell us the matter
density equals the summed spacetime curvature. So, if the matter density is
infinite, the curvature of spacetime has become
infinite as well.

That last statement cannot be literally correct. According to
Einstein's general theory of relativity, spacetime at every event has definite
curvature. If that curvature is everywhere infinite, we define no spacetime at
all. If we try to imagine the time of the big bang itself as one of the times
of the cosmology, we are saying that there is a time at which spacetime is not
properly defined. So there can be no time in the cosmology corresponding to the
big bang. We describe the big bang as a "singularity," a breakdown in the laws that govern space
and time.

The term
singularity, roughly speaking, designates a point in a mathematical
structure where a quantity fails to be well defined, even though the quantity
is well defined at all neighboring points. The simplest and best known example
arises with the inverse function, 1/x. As long as x is non-zero, 1/x is well
defined. For

The system has a singularity at x=0, for then 1/x = 1/0 and, as we
all learn in our arithmetic classes, "you cannot divide by zero." There is a
temptation to say that 1/0 is "infinity." But that is dangerous. As we have
just seen, if we approach x=0 from positive values of x, the inverse 1/x grows
without limit towards +infinity (i.e. "plus infinity"). If we approach x=0 from
negative values of x, the inverse 1/x grows negatively without limit towards
-infinity (i.e. "minus infinity"). If we insist on giving 1/0 a value, which do
we give? "Plus-infinity" or "minus-infinity"? The safer course is just to say
that we have a singularity at x=0 and not try to give it any value.

What we can say is this. The universe has an age or time--its age after the big bang. The spacetime of
the universe exists for every age greater than zero: 1 million years, one
hundred years, one second, one half second, one tenth second, and so on. No
matter how small we make the age, there is a corresponding spacetime, as long
as the age is greater than zero. But nothing in general relativity--no spacetime at all--corresponds to the zero age.

This moment of zero age is a fictitious moment in the history of the universe. In that
regard, it is like the fictitious point "at infinity" on the horizon where
parallel lines meet. Of course well all know that there is no such point,
although we see it drawn routinely in perspective drawings.

Cosmological Red Shift

We can now return to the red shift that figures in the Hubble
expansion and give a more precise account of its origin. It is not a
traditional Doppler shift, but something more subtle. A distant galaxy emits
light towards us. The light waves with their crests are carried by space
towards us. For a distant galaxy, it can take a very long time for the light to
reach us. During that time, the cosmic expansion of
space proceeds. The effect is that the waves of the light signal get
stretched with space. So the wavelength of the light increases and its
frequency decreases. It becomes red shifted.

To get a sense of the process, imagine a column of ants setting off to walk across a rubber sheet.
They may enter the sheet at a rate of one ant per second. If the rubber sheet
is stretched while the ants walk, each ant will need to go further to get to
the other side than the one before. So the ants will arrive less frequently at
the other side than the original rate of one ant per second.

Cosmic Dynamics: Three Possibilities

What is the overall dynamics of spacetime? Einstein's
gravitational field equations applied to the Friedmann-Robertson-Walker
spacetimes give us three possibilities, cataloged
below as I, II or III.

What decides between them is the density of matter. The
so-called "critical density" of matter is the
deciding value. It is a minute average density of 10-29grams per
cubic centimeter. Our cosmology will be one of the three shown in the table
below according to whether the actual average density of matter in our universe
is greater than, equal to or less than this critical density.

Cosmology

I

II

III

Average mass density

Greater than critical

Critical

Less than critical

Geometry of space

Spherical
positive curvature

Flat, Euclidean
zero curvature

Hyperbolic
negative curvature

Dynamics

Expands and collapses to big crunch

Expands indefinitely

Expands indefinitely

The table gives the broad features. In cases I and III, space is curved. The
scale factor R--the radius of curvature of the
space--determines the extent of curvature. (The radius of curvature of a three
dimensional space is the three-dimensional analog of the radius of a
two-dimensional sphere.) The value of R differs greatly according to the
particular matter density at hand. However a rough estimate is this:

Scale
factor
R

very
roughly
equals

Hubble
age of
universe

x

speed
of light

So by this estimate the scale factor is roughly 14 billion light years. This value only obtains exactly
for special cases. In cosmologies I, it obtains exactly if the average matter
density is twice the critical.

We can also get a sense of the dynamics by plotting how the scale factor R changes with time in typical
examples of the three cosmologies. In the case of cosmologies II with Euclidean
geometry, the scale factor R is simply set to be the distance between two
conveniently placed galaxies. As the cosmic expansion proceeds, R grows in
response.

In general there are no simple
formulae for these curves. One case proves to be simple. In Cosmologies
II, if all the matter is what is called "dust" in the jargon (i.e. ordinary
matter like our earth), then R increases in direct proportion with
(time)2/3. Or in that cosmology, if all the matter is radiation, R
increases in direct proportion with (time)1/2.

A Newtonian Analogy for the Dynamics

At first the dynamics seems arbitrary. Why should the different universes have the properties
they do? Why, for example, should a universe with greater mass density only
have a big crunch? And why with lesser mass density, will the expansion
continue indefinitely? We can makes some sense of this with an analogy from
Newtonian theory.

There is a reason Newtonian theory can tell us something.
Recall that general relativity turns back into
Newtonian theory as long as we consider ordinary conditions: nothing moves
quickly, there are no strong gravitational fields and--most important here--we
consider small distances, not cosmic distances.

So it turns out that a tiny chunk of the cosmic fluid of a
Friedmann-Robertson-Walker spacetime is governed by Newtonian principles. The
easiest way to see those principles in action is to consider a closely analogous system in Newtonian theory.

Imagine that we have a bomb in
space that explodes. It will spread debris into space. Each fragment in
the debris will attract all the others according to Newton's inverse square law
of gravity. The ultimate fate of the debris cloud depends on the balancing of
the initial magnitude of the explosion with the strength of the gravitational
attraction within the debris cloud.

If there is a greater amount of
matter in the original lump, the explosion will produce a denser cloud
of debris. Its internal forces of gravitational attraction will be strong
enough to slow and halt the initial outward motion of the explosion and draw
the fragments back together, bringing about a collapse. It corresponds to the
dynamics of cosmology I; there is a big bang and a big crunch.

If there is a lesser amount of
matter in the original lump, the explosion will produce a more dilute
debris cloud whose internal forces of attraction will not be sufficient to halt
the initial outward motion of the blast. That outward motion will continue
indefinitely. It corresponds to cosmology III; there is a big bang and
no big crunch.

We could imagine an intermediate
case in which the explosion is just energetic enough to fling the debris out of
the reach of the gravitational forces; any weakening of the explosion would be
too weak to prevent collapse. This corresponds to the intermediate case of
cosmology II.

The Newtonian analogy is useful in so far as it gives us a
nice picture for the dynamics. But it omits a lot.
There is no account of the different spatial geometries and the big bang is the
explosion of a nugget of matter into a pre-existing space. That is not what is
portrayed by relativistic cosmologies.

And Ours Is...

Which of these three cosmologies
is ours? The questions is not without some interest. If it is the first
cosmology I, then we live in a finite space. If we point in in any direction,
after some finite distance, we are pointing at the back of our own heads!
Further, just as the universe has a finite past bounded by the big bang, so
there is also an end in our future. The entire universe will collapse down onto
itself in a "big crunch". In cosmologies II and III neither of these results
obtain. Cosmology II, however, is the only one in which the geometry of space
on cosmic scales is Euclidean.

The value of the critical density
is extremely small: 10-29grams per cubic centimeter of space. That
is 0.00000000000000000000000000001 grams per cubic centimeter. That is very
little indeed! It corresponds to roughly 5 hydrogen atoms only in a cubic meter
of space. That sort of vacuum is extremely hard to achieve with laboratory
equipment on earth.

Here's another measure of how small it is. Take
one fifth of a teaspoon of water, which is
roughly 20 drops. (That amounts to one gram.) How widely spread must it
be in order to match the critical density? Guess! What if we take those
20 drops and spread them over the volume of the Astrodome? Not even
close. Think bigger. What about 20 drops spread over the volume of
earth? Better, but still too small. That density is still 100 times too
big. Those 20 drops of water need to spread over one hundred earth
volumes if their dilution is to match the critical density!That, at least, is what my sums show. The
radius of the earth is about 6,366,000 meters. So its volume is 1.08 x
1021 m3, which comes to 1.08 x 1027
cubic centimeters. So one gram spread over this volume is still roughly
100 times too dense.

Since this critical density is so small, you might think that
our universe must have an average density more than critical. That would be
jumping to conclusions. What counts is the density of matter averaged over all space. So we need to take the
matter of earth and spread it over the vast emptiness of space between stars
and galaxies. And then the calculation gets more complicated because of the
steady accumulation of evidence that a very substantial portion of the energy
of our universe is "dark," so its existence is actually inferred indirectly
from the gravitational effects it produces.

The upshot is that the average density of matter comes out
very close to the critical. Indeed the astonishing
and maddening result is that the more accurately it is measured, the closer our
density gets to the critical value. So we remain unable to say which of the
cosmic scenarios above is our own.

The suspicion is growing that our density may be exactly the
critical density. It seems too much of a
coincidence that of all values our matter density could have, it just
turns out to be so close to the critical density. So the supposition is that
there might be some cosmic process that has driven the matter density to this
value. So called "inflationary" cosmologies posit an early phase of very rapid
cosmic expansion that would have the effect of driving the matter density
towards the critical.

What you should know

How Friedmann-Robertson-Walker spacetimes form the basis of modern big
bang cosmology.

The three types of universes in FRW cosmology and what decides between
them.

The Newtonian analogs for big bang cosmology.

That the big bang is not a moment in time.

Copyright John D. Norton. March 2001; January 2007,
February 16, 23, October 16, November 10, 2008, March 31, 2010; January 1, 2013; March 18, 2015.